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a really tiny lemma, but I gave it an entry such as to have an easier time pointing to it: closed injections are embeddings
stub for embedding of smooth manifolds
Created closed map to satisfy a few links. I notice that there’s quite a lot at open map. Does any of the more general picture carry over to closed maps?
wrote out statement and proof at
open subspaces of compact Hausdorff spaces are locally compact
and referenced this where the claim is stated at locally compact Hausdorff space.
I have edited at Tychonoff theorem:
tidied up the Idea-section. (Previously there was a long paragraph on the spelling of the theorem before the content of the theorem was even mentioned)
moved the proofs into a subsection “Proofs”, and added a pointer to an elementary proof of the finitary version, here
Notice that there is an ancient query box in the entry, with discussion between Todd and Toby. It would be good to remove this box and turn whatever conclusion was reached into a proper part of the entry.
At then end of the entry there is a line:
More details to appear at Tychonoff theorem for locales
which however has not “appeared” yet.
But since the page is not called “Tychonoff theorem for topological spaces”, and since it already talks about locales a fair bit in the Idea section, I suggest to remove that line and to simply add all discussion of localic Tychonoff to this same entry.
at separation axiom I have expanded the Idea section here, trying to make it more introductory and expository.
I saw tube lemma, and decided to bulk it up.
Many books (such as the famous topology text by Munkres) give proofs which involves multiple subscripts and multiple choices; I’ve written arguments to mostly eliminate that.
we have a page semi-locally simply connected space but no page locally simply connected space. I am not sure I know the difference! And there is no mention of it in the entry that we have. How is this resolved?
Some nLab pages had a gray link to descriptive set theory, which now has the following stub:
Descriptive set theory is the study of the structures and hierarchies of subsets of real numbers (or more generally of subsets of Polish spaces) that are definable by formulas with real parameters in second-order arithmetic.
Such subsets include Borel sets and more generally projective sets that are defined by alternating between taking images under projection maps of previously defined sets and taking complements of previously defined sets. Once the domain of topologists of the Polish schools and Russian analysts of the early 20th century, it is now considered a central area of logic in which set theory and computability theory (recursion theory) meet and interact.
I have written an article closed-projection characterization of compactness, so as to record a nice way of proving the Tychonoff theorem that is due to Clementino and Tholen. It’s rather direct and elementary, which doesn’t involve ultrafilters or nets or any such machinery. This might make it a possibility for a strong undergraduate classroom. (Munkres also has a proof which I haven’t cross-checked; the one I wrote up involves a smidge of categorical terminology, notably inverse limits.)
I didn’t want to stick in the proof at Tychonoff theorem as that article might be getting a bit bloated, but I did link there to the new article.
I created Cantor space to record its definition as a locale, but goodness knows there is no end to what might be written about it.
I noticed that presently topological basis redirects to basis in functional analysis instead of to the entry topological base. This seems dangerous. I’d like to change that redirect.
At limits and colimits by example an entry for Kuratowski pair was requested, so I created it. The place where this should really be linked to is the entry ordered pair, so I added a link there. Also, I made that entry mention Cartesian product, which it didn’t.
I have given product topological space an semi-informal Idea-section that means to quickly and transparently tell the reader what they need to know.
This is in reaction to what we presently have at Tychonoff product which presently seems needlesly intransparent for the purpose of a reader who just wants to know what the open subsets actually are.
I was thinking about merging product topological space with Tychonoff product. I haven”t yet, but mostly just due to lack of energy.
The entry for infinitesimal extension said that an infinitesimal extension of rings was an epimorphism of rings with nilpotent kernel. I’ve changed this to say a quotient map of rings with nilpotent kernel. I hope this is correct: for example, localization maps are ring epimorphisms, and often have zero kernel (so in particular, nilpotent kernel) but geometrically these correspond to dense open inclusions, which are in no sense infinitesimal extensions.
Finally created stereographic projection. Lots more could be said, including metric/conformal aspects.
Somebody anonymous has created codiagonal comodule structure with no content. But let’s leave the it there and simply fill in some.
I was going to start game semantics to record a couple of references to dependent type theory, but I’m getting an error message at the moment. So I’ll just leave here for now:
In logic, game semantics is used to provide a semantic interpretation of logic constructions in terms of strategies for opposing players to win a game corresponding to some proposition.
For attempts to formulate a game semantics for dependent type theory, see
Matthijs Vakar, Radha Jagadeesan, Samson Abramsky, Game Semantics for Dependent Types, (pdf)
Norihiro Yamada, _Game Semantics for Martin-Löf Type Theory, (arXiv:1610.01669)
Presently I am concerned with the following: I want to teach some basics of limits and colimits of topological spaces to undergraduates. I had tried to gently introduce some category theoretic terminology as I introduced topological spaces as such, but a little testing reveals that part of the audience would rather not see these side remarks turn out to become required reading.
But now since all the shapes of diagrams that I’d be about to consider anyway are free, I figured I circumvent the need to speak of diagrams as functors by restricting to free domains and simply giving everything explicitly in components, with the underlying category theory again relegated to side-remarks that may be ignored at will.
I am trying my hands at an exposition of this kind in a new entry free diagram.
Eventually this kind of material might also be worthwhile as introductory exposition at limits and colimits by example.
since this was missing, I created a minimum at equivalence in a 2-category
I added a little bit to maximal ideal (first, a first-order definition good for commutative rings, and second a remark on the notion of scheme, adding to what Urs wrote about closed points).
The second bit is almost a question to myself: I don’t feel I really grok the notion of scheme (why it’s this and not something slightly different that’s the natural definition, the Tao if you like). In particular, it’s where fields – simple objects in the category of commutative rings – make their entrance in the notion of covering by affine opens that I don’t feel I really understand.
I have a request to the logic experts:
The entry classical logic is a bit thin. I would like to be able to point to it so that readers who don’t already know about it all, get away with a decent idea of what is meant. Might somebody have the energy to add a few lines?
Of course I could add a few lines myself, but I imagine it would be more efficient if some expert here did it from scratch, instead of having to improve on what I would come up with.
Especially it would be nice if the following keywords were at least mentioned and maybe briefly commented on in the entry:
Also for instance the keyword constructive anything is presently missing from the entry.
Thanks!
had occasion to give attaching space its own little entry. Cross-linked with Top and CW-complex.
added section on Russell’s relation to mysticism based on his essay Mysticism and Logic and the biography of Ray Monk.
I added a sentence on ’eigenfunction’ to eigenvector.
Wrote about Conduche functors.
I have created an entry
to be used for inclusion in other entries, and I have included it into relevant entries (mostly in their subsections “Related entries”, except at Top and at limits and colimits by example).
it seems we were lacking order topology, so I created a minimum and cross-linked a bit.
partition of unity, locally finite cover
Will put up some stuff about Dold’s trick of taking a not-necessarily point finite partition of unity and making a partition of unity. There is a case when I know it works and a case I’m really not sure about - I need to find where the argument falls down because I get too strong a result. I’ll discuss this in the thread soon, and port it over when it is stable.
I have touched the entry K-topology. Polished the definition and spelled out in some detail why it is not regular (while clearly Hausdorff).
The example which used to be in the entry (rational numbers with their subspace K-topology here) ends with
This space can be used to construct a quasitopological groupoid which isn’t a topological groupoid.
This statement should be accompanied with some reference. I suppose it refers to a construction that David R. (who wrote this back in 2010, rev #1) used in one of his articles?
I am splitting off Zariski topology from Zariski site, in order to have a page for just the concept in topological spaces.
So far I have spelled out the details of the old definition of the Zariski topology on (here).
Wrote out a proof for paracompact Hausdorff spaces are normal.
(By the way, I also looked at TopoSpaces here to check what they offer, and am a bit dubious about their step 5. But maybe I am misreading it. In any case, I feel there is a simpler way to state the proof.)
I have added to Galois connection some more remarks to the Idea section, and expanded the Examples-section with the material that Todd wrote here.
started a stub for topological invariance of dimension
found it necessary to split off geometric realization of categories as a separate entry, recorded Quillen’s theorems A and B there
all very briefly. I notice that David Roberts has more on his personal web (have included it as a reference)
at schemes are sober I have added pointer to a comprehensive proof, here
I have added to alternative algebra the characterization in terms of skew-symmetry of the associator.
gave finite intersection property an entry, added statement and proof of the corresponding characterization of compact topological spaces. Pointed to this from Tychonoff theorem, where the statement is used.
I added recollement as an example to the pages on reflective and coreflective subcategories.
I did a little editing over at empty set; the query-box discussion of looked like it could be summarized with dispatch and relegated to a remark. Revert back or re-edit if you don’t like it.
added to Hausdorff topological space a brief paragraph Beyond topological spaces
I have substantially expanded the exposition at diagram. In the section Diagrams shaped like categories.
First I expanded on the quick functorial definition of diagrams and their ((co-)limiting) (co-cones) and then I added a detailed component-based definition meant for readers not fluent in unwinding the former definition.
Haven’t proof-read yet, as my battery is dying now.
I noticed that the entry disjoint union had no pointer to or from either of dependent sum, dependent sum type.
I have added the bare pointers now (no energy for more now). But this means some basic examples are missing in these entries.
at triangle inequality the discussion of the interpretation in enriched category theory had been missing. I have added in a corrresponding section here and cross-linked with Lawvere metric spaces.
added to interior the relation between topological interior and topological closure, here (also at topological closure, here)
I was un-graying some links at epsilontic analysis. Among the titles of non-existing entries that are still being requested is
“classical analysis”
“topological property”.
Is it likely/desireable that we will have entries with these titles? Maybe we should change these links to point to “analysis” and to “topology” instead?
started sidebar topology - contents, but am running out of time for this…
For completeness I created Kolmogorov space.
added statement and proof at compact Hausdorff spaces are normal.
Also added a pointer to this entry from the entries that previously claimed this without reference.
I started an entry for Rijke’s join of maps. How do you place the lower?
I have added both to proof and to experiment pointer to
with the quote (from p. 2):
we claim that the role of rigorous proof in mathematics is functionally analogous to the role of experiment in the natural sciences
added statement and proof that sequentially compact metric spaces are equivalently compact metric spaces
added statement and proof at continuous images of compact spaces are compact. Cross-linked with extreme value theorem and other related entries.
added a proof at compact subspaces of Hausdorff spaces are closed
added statement-entry closed subspaces of compact Hausdorff spaces are equivalently compact subspaces.
with proof by pointing to the two directions:
Cross-linked with compact space, closed subspace and also with Heine-Borel theorem
created the line with two origins
I added a remark about the contravariant Yoneda embedding to the page on the Yoneda embedding.
It’s pretty elementary, but I think worth mentioning for those new to category theory that this is just the Yoneda embedding of the opposite category .
added statement and proof that sequentially compact metric spaces are totally bounded
added statement and proof to Lebesgue number lemma.
did the same for sequentially compact metric spaces are equivalently compact metric spaces but now the Lab seems to have gone down before I could submit the edit.